梯度映射的离散性与开性的Pogorelov型反例
Critical Sobolev thresholds for openness and discreteness of Monge-Ampere gradient mappings
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中文总结 AI 辅助
本文针对Guerra-Tione提出的梯度映射是否必为开且离散的问题，通过Pogorelov型构造证明n≥4时答案为否，且该构造在n=3时无法直接解决三维情形。
中文摘要 AI 辅助
设Ω是ℝⁿ中的区域，假设u属于W²,ⁿ_loc(Ω)，且在Ω中几乎处处满足det D²u≥δ>0。Guerra和Tione提出的问题是：梯度映射Du是否一定是开且离散的？本文给出了一个明确的Pogorelov型构造，证明在所有n≥4的维度中答案是否定的：存在u∈W²,ⁿ_loc(Ω)，满足上述Hessian行列式下界，且几乎处处有D²u>0，使得Du将整个线段坍缩为单点，因此不是离散的。我们还证明，当n=3时，同一构造存在对数发散，因此无法直接解决三维情形的问题。
英文摘要
Let $Ω\subset\R^n$ be a domain and let $u\in W^{2,n}_{\loc}(Ω)$ satisfy \[ \det D^2u\geqδ>0\qquad\text{a.e. in }Ω. \] Guerra and Tione asked whether the gradient mapping $Du$ must be open and discrete. We give a negative answer in every dimension $n\geq4$, in a form that isolates the sharp regularity mechanism. First, a local Pogorelov model is chosen to solve the exact equation $\det D^2u=1$ a.e. It is convex, belongs to $C^{1,1-2/n}\cap W^{2,n}_{\loc}$, and its Hessian is positive definite off a line. Nevertheless, its gradient collapses that line to one point. In fact the gradient is neither open nor discrete, its branch set is exactly the collapsed line, and its Jacobian equals one a.e. We then develop a $k$-dimensional version of the construction. If $1\leq k<n/2$, there are convex potentials with a $k$-dimensional flat contact set, uniformly positive and bounded Hessian determinant, and exact critical exponent \[ p_{n,k}=\frac{n(n-k)}{2k}. \] Their Hessians belong to $L^p_{\loc}$ precisely for $p<p_{n,k}$, lie in the weak endpoint space $L^{p_{n,k},\infty}_{\loc}$, and fail to belong to any finite-index Lorentz endpoint. This matches the critical minimum-set theorem of Collins and Mooney. At the regularity required in the question, the construction produces branch sets of every integer dimension $k<n/3$. We also give a topological reformulation of the problem. In the convex branch, gradient fibers are exactly contact sets with supporting affine functions, and openness and discreteness are equivalent to strict convexity. For a general gradient under the hypotheses above, critical Sobolev mapping theory already supplies continuity and sense preservation. The question asks whether the monotone factor in the Eilenberg--Whyburn monotone--light factorization is trivial.
发表机构
- Institute of Applied Mathematics, Shenzhen Polytechnic University（深圳职业技术大学应用数学研究所）
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